State Estimation. Ali Abur Northeastern University, USA. September 28, 2016 Fall 2016 CURENT Course Lecture Notes

Transcription

1 State Estmaton Al Abur Northeastern Unversty, USA September 8, 06 Fall 06 CURENT Course Lecture Notes

2 Operatng States of a Power System Al Abur NORMAL STATE SECURE or INSECURE RESTORATIVE STATE EMERGENCY STATE PARTIAL OR TOTAL BLACKOUT OPERATIONAL LIMITS ARE VIOLATED

3 Energy Management System Applcatons SCADA / EMS Confguraton Al Abur Measurements State Estmaton Topology Processor Securty Montorng Load Forecastng External Equvalents Emergency Control Restoratve Control Contngency Analyss On-lne Power Flow Secure Y STOP N Securty Constraned OPF Preventve Acton

4 State Estmaton and Related Functons Weghted Least Squares (WLS) Estmator Al Abur Analog Measurements P, Q, P f, Q f, V, I, θ k, δ k Topology Processor State Estmator (WLS) V, θ Bad Data Processor Load Forecasts Generaton Schedules Network Observablty Analyss Pseudo Measurements [ njectons: P, Q ] Crcut Breaker Status Assumed or Montored

5 Power Flow and State Estmaton Comparson Al Abur Power Flow State Estmaton Termnaton crteron Bus P,Q msmatch ΔX State ncrement Formulaton Determnstc Stochastc Soluton Depends on choce of slack bus Independent of reference bus selecton Bus Types Important Irrelevant Loads Modeled Not used Generator Lmts Modeled Not used Transmsson System Modeled Modeled

6 Power System State Estmaton Problem Statement Al Abur [] : Measurements P-Q njectons P-Q flows V magntude, I magntude [x] : States V, θ, Taps (parameters) EXAMPLE: [] [ P; P3; P3; P; P; P3; V; Q; Q3; Q3; Q; Q; Q3 ] m 3 (no. of measurements) [x] [ V; V; V3; θ; θ3 ] n 5 (no. of states)

7 Network Model Bus/branch and bus/breaker Models Al Abur Bus/Breaker Bus/Branch Topology Processor

8 Measurements Bus/branch and bus/breaker Models Al Abur Bus/branch Bus/Breaker V

9 Measurement Model [ m ] [h([x])] + [e] Al Abur +e +e State Estmator 3 +e 3 : true measurement e : measurement error e e s + e r systematc random

10 Measurement Model Al Abur Assumptons e ~ N ( 0, σ ) olds true f: e s 0, e r ~ N ( 0, σ ) If e s 0, then E(e ) 0,.e. SE wll be based!

11 Maxmum Lkelhood Estmator (MLE) Lkelhood Functon Al Abur Consder the random varables X, X,, X n wth a p.d.f of f(x θ), where θ s unknown. The jont p.d.f of a set of random observatons x { x, x,, x n } wll be expressed as: f n ( x θ) f (x θ) f(x θ) f(x n θ ) Ths jont p.d.f s referred to as the Lkelhood Functon. The value of θ, whch wll maxme the functon fn( x θ) wll be called the Maxmum Lkelhood Estmator (MLE) of θ.

12 Maxmum Lkelhood Estmator (MLE) Maxmum Lkelhood Estmator Al Abur } ) ( exp{ ) ( σ µ πσ f ) ( ) ( ) ( ) ( m m m m m f f f f m m m m m f f L log log ) ( ) ( log ) ( log σ π σ µ Normal (Gaussan) Densty Functon, f() Lkelhood Functon, f m () Log-Lkelhood Functon, L

13 Maxmum Lkelhood Estmator (MLE) Weghted Least Squares (WLS) Estmator Al Abur Gven the set of observatons,,, n MLE wll be the soluton to the followng: Mnme Subject to m W h r Maxme OR Mnme ( x) + r f m m ( ) µ σ ( ) Defnng a new varable r, measurement resdual:,.., m W σ µ E( ) h ( x) The soluton of the above optmaton problem s called the weghted least squares (WLS) estmator for x.

14 Maxmum Lkelhood Estmator (MLE) Weghted Least Squares (WLS) Estmator Al Abur ] [ ] [ ] [ ] Subject to [ Mnme r x W r m + } { W dag W W σ Lnear case: Soluton s gven by: ] [ ] [ ] [ ] [ [x] W G T ] [ ] [ ] [ ] [ W G T

15 Measurement Model Al Abur Gven a set of measurements, [] and the correct network topology/parameters: [] [h ([x]) ] + [e] Measurements: Contan errors True System States: Unknown! Measurement Errors: Unknown!

16 Measurement Model Al Abur Followng the state estmaton, the estmated state wll be denoted by [ xˆ ]: [] [h ([ xˆ ]) ] + [r] Measurements: Contan errors Estmated System States Measurement Resduals: Computed

17 Smple Example 3 4 Z h θ + e Z.0 / 0.0 / θ 4 3 r r 3 r 4 SLOPEθ* r h h h 3 h 4 h : ESTIMATED MEASUREMENT : MEASURED VALUE r : MEASUREMENT RESIDUAL Z h θ* Al Abur

18 Weghted Least Squares (WLS) Estmaton Al Abur Mnme r + ωr + ω3r3 ω4r4 ω + What are weghts, w? ω.0 σ ow are they chosen? σ Assumed error varance of measurement.

19 Network Observablty Defntons Al Abur Fully observable network: A power system s sad to be fully observable f voltage phasors at all system buses can be unquely estmated usng the avalable measurements.

20 Network Observablty Necessary and Suffcent Condtons Al Abur n mn m n n m p Z θ θ θ 3 3 m n NECESSARY BUT NOT SUFFICIENT EXAMPLE: m, n, UNOBSERVABLE SYSTEM Rank() n SUFFICIENT [ 3 ] θ θ State Vector ] [ ] [ ] [ ] [ ˆ P T Z W G θ Sngular Matrx! Cannot be nverted.

21 Measurement Classfcaton Types of Measurements Al Abur. CRITICAL MEASUREMENTS If they are lost or temporarly unavalable, the system wll no longer be observable, thus state estmaton can not be executed If they have gross errors, they can not be detected Measurement resduals wll always be equal to ero,.e. crtcal measurements wll be perfectly satsfed by the estmated state. REDUNDANT MEASUREMENTS CAN BE REMOVED WITOUT AFFECTING NETWORK OBSERVABILITY

22 Network Observablty Defntons Al Abur Unobservable branch: If the system s found not to be observable, t wll mply that there are unobservable branches whose power flows can not be determned. Observable sland: Unobservable branches connect observable slands of an unobservable system. State of each observable sland can be estmated usng any one of the buses n that sland as the reference bus.

23 Network Observablty Defntons Al Abur Observable Islands RED LINES: Unobservable Branches

24 Mergng Observable Islands Pseudo-measurements Al Abur If the system s found unobservable, use pseudo-measurements n order to merge observable slands. Pseudo-measurements: Forecasted bus loads Scheduled generaton Select pseudo-measurements such that they are crtcal. Errors n crtcal measurements do not propagate to the resduals of the other (redundant) measurements.

25 Observable Islands Unobservable Branches Al Abur ISLAND ISLAND ISLAND 3

26 Robust (reslent) Estmaton Reslency: A Smart Grd Requrement Al Abur If an estmator remans nsenstve to a fnte number of errors n the measurements, then t s consdered to be robust. Example: Gven { 0.9, 0.95,.05,.07,.09 }, estmate usng the followng estmators:.. Xˆ Xˆ mean{ } 5 medan{ }, Soluton: Replace 5.09 by an nfntely large number 5. 5 The new estmate wll then be: ˆ 5 Ths estmator s NOT robust. Replace both 5 and 4 by nfnty. a b The new estmate wll then be: Ths s a more robust estmator than the one above. 5,...,5 X a X ˆ b.05 (fnte)

27 Robust Estmaton M-Estmators Al Abur M-Estmators (uber 964) Consder the problem: Mnme Subject to m ρ( r ) h( x) + r Where ρ( r ) s a chosen functon of the measurement resdual In the specal case of the WLS state estmaton: ρ ( r ) r σ

28 Robust Estmaton M-Estmators Al Abur Some Examples of M-Estmators otherwse ) ( a a r r r σ σ σ ρ otherwse ) ( a r a a r r r σ σ σ σ ρ Quadratc-Constant Quadratc-Lnear r r ) ρ( Least Absolute Value (LAV)

29 Robust Estmaton LAV Estmator Example Al Abur A Measurement Model: + A x + e,..., 5 x Measurements: Z A A LAV estmate for x and measurement resduals: x r T T [3.005; 4.00] [ 0.0; 0.05; 0.0; 0.0; 0.0] CANGE measurement 5 from 5.0 to 5.0 ( Smulated Bad Datum ): LAV estmate for x and measurement resduals: x r T T [3.0; 4.0] [ 0.0; 0.0; 0.045; 0.0; 9.98 ]

30 Robust Estmaton LAV Estmator Example Al Abur A Measurement Model: + A x + e,..., 5 x Measurements: Z A A LAV estmate for x and measurement resduals: x r T T [3.005; 4.00] [ 0.0; 0.05; 0.0; 0.0; 0.0] CANGE measurement 5 from 5.0 to 5.0 ( Smulated Bad Datum ): LAV estmate for x and measurement resduals: x r T T [3.0; 4.0] [ 0.0; 0.0; 0.045; 0.0; 9.98 ]

31 Bad Data Detecton Ch-squares χ Test Al Abur Consder X, X, X N, a set of N ndependent random varables where: X ~ N(0,) Then, a new random varable Y wll have a freedom,.e.: χ N X Y ~ χ N dstrbuton wth N degrees of

32 Bad Data Detecton Al Abur Now, consder the functon f ( x) and assumng: e m N R e N m ~ r ~ N e R (0,) m ( N e ) f(x) wll have a χ dstrbuton wth at most (m-n) degrees of freedom. In a power system, snce at least n measurements wll have to satsfy the power balance equatons, at most (m-n) of the measurement errors wll be lnearly ndependent.

33 Bad Data Detecton Detecton Algorthm χ --Test Al Abur Solve the WLS estmaton problem and compute the objectve functon: m ( h ( x)) Look up the value correspondng to p (e.g. 95 %) probablty and (m-n) degrees of freedom, from the Ch-squares dstrbuton table. Let ths value be J ( x) χ ( m n), p σ ere: p Pr{ J ( x) χ ( m n), p } Test f J ( x) χ( m n), p If yes, then bad data are detected. Else, the measurements are not suspected to contan bad data.

34 Bad Data Identfcaton Propertes of Measurement Resduals Al Abur Lnear measurement model: K s called the hat matrx. Now, the measurement resduals can be expressed as follows: where S s called the resdual senstvty matrx. ) (, ˆ ˆ R R K K x T T R R x T T ) ( ˆ Se e K I e x K I K I r + ] [Note that K ) ( ) )( ( ) ( ˆ

35 Bad Data Identfcaton Dstrbuton of Measurement Resduals Al Abur The resdual covarance matrx Ω can be wrtten as: E[ rr S T ] Ω S E[ e e ] S R S T S ence, the normaled value of the resdual for measurement wll be gven by: R N r r Ω The row/column of S correspondng to a crtcal measurement wll be ero. If there s a sngle error n the measurement set (provded that t s not a crtcal measurement) the largest normaled resdual wll correspond to that error. R r T S T

36 Bad Data Identfcaton Largest Normaled Resdual Test Al Abur. Compute the normaled resduals. Fnd k such that r N k s the largest among all r N,,,m.. 3. If r N k > c3.0, then the k-th measurement wll be suspected as bad data. Else, stop, no bad data wll be suspected. 4. Elmnate the k-th measurement from the measurement set and go to step.

37 Use of Synchrophasor Measurements Al Abur Gven enough phasor measurements, state estmaton problem wll become LINEAR, thus can be solved drectly wthout teratons teratve Non Z R R X e X Z Measurements Phasor Iteratve Z R R X e X h Z Measurements Conventonal T T + + ) ( ˆ ) ( ˆ ) (

38 Placng PMUs: Al Abur : Power Injecton : PMU : Power Flow : Voltage Magntude

39 Explotng ero njectons Al Abur : Power Injecton : PMU : Power Flow : Voltage Magntude

40 Use of Synchrophasor Measurements Al Abur Gven at least one phasor measurement, there wll be no need to use a reference bus n the problem formulaton Gven unlmted number of avalable channels per PMU, t s suffcent to place PMUs at roughly /3 rd of the system buses to make the entre system observable just by PMUs. Systems No. of ero njectons Number of PMUs Ignorng ero Injectons Usng ero njectons 4-bus bus bus 0 3 9

41 Performance Metrcs Al Abur State Estmaton Soluton Accuracy: Varance of State nverse of the gan matrx, [G] - E[ (x x*) (x x*) ] Convergence: Condton Number Rato of the largest to smallest egenvalue Large condton number mples an ll-condtoned problem.

42 Performance Metrcs Al Abur Measurement Desgn Crtcal Measurements: Number of crtcal measurements and ther types Local Redundancy Number of measurements ncdent to a gven bus (N-) Robustness Capablty of the measurement confguraton to render a fully observable system durng sngle measurement and branch losses

43 Summary Al Abur State Estmaton and ts related functons are revewed. Importance of measurement desgn s llustrated. Commonly used methods of dentfyng and elmnatng bad data are descrbed. Impact of ncorporatng phasor measurements on state estmaton s brefly revewed. Metrcs for state estmaton soluton and measurement desgn are suggested.

44 Power Educaton Toolbox (P.E.T) Power Flow and State Estmaton Functons Al Abur Free software to: Buld one-lne dagrams of power networks Run power flow studes Run state estmaton

45 References Al Abur F.C. Schweppe and J. Wldes, ``Power System Statc-State Estmaton, Part I: Exact Model'', IEEE Transactons on Power Apparatus and Systems, Vol.PAS-89, January 970, pp.0-5. F.C. Schweppe and D.B. Rom, ``Power System Statc-State Estmaton, Part II: Approxmate Model'', IEEE Transactons on Power Apparatus and Systems, Vol.PAS-89, January 970, pp F.C. Schweppe, ``Power System Statc-State Estmaton, Part III: Implementaton'', IEEE Transactons on Power Apparatus and Systems, Vol.PAS-89, January 970, pp A. Montcell and A. Garca, "Fast Decoupled State Estmators", IEEE Transactons on Power Systems, Vol.5, No., pp , May 990. A. Montcell and F.F. Wu, ``Network Observablty: Theory'', IEEE Transactons on PAS, Vol.PAS-04, No.5, May 985, pp

46 References Al Abur A. Montcell and F.F. Wu, ``Network Observablty: Identfcaton of Observable Islands and Measurement Placement'', IEEE Transactons on PAS, Vol.PAS-04, No.5, May 985, pp G.R. Krumphol, K.A. Clements and P.W. Davs, ``Power System Observablty: A Practcal Algorthm Usng Network Topology'', IEEE Trans. on Power Apparatus and Systems, Vol. PAS-99, No.4, July/Aug. 980, pp A. Garca, A. Montcell and P. Abreu, ``Fast Decoupled State Estmaton and Bad Data Processng'', IEEE Trans. on Power Apparatus and Systems, Vol. PAS-98, pp , September 979. Xu Be, Yeojun Yoon and A. Abur, Optmal Placement and Utlaton of Phasor Measurements for State Estmaton, 5th Power Systems Computaton Conference Lège (Belgum), August -6, 005.